Algebra Examples

$y = \ln (x + \sqrt{x^{2} + 1})$

Step 1

Rewrite the equation as $\ln (x + \sqrt{x^{2} + 1}) = y$ .

$\ln (x + \sqrt{x^{2} + 1}) = y$

Step 2

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x + \sqrt{x^{2} + 1})} = e^{y}$

Step 3

Rewrite $\ln (x + \sqrt{x^{2} + 1}) = y$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{y} = x + \sqrt{x^{2} + 1}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $x + \sqrt{x^{2} + 1} = e^{y}$ .

$x + \sqrt{x^{2} + 1} = e^{y}$

Step 4.2

Subtract $x$ from both sides of the equation.

$\sqrt{x^{2} + 1} = e^{y} - x$

Step 4.3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{2} + 1}}^{2} = {(e^{y} - x)}^{2}$

Step 4.4

Simplify each side of the equation.

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Step 4.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 1}$ as ${(x^{2} + 1)}^{\frac{1}{2}}$ .

${({(x^{2} + 1)}^{\frac{1}{2}})}^{2} = {(e^{y} - x)}^{2}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Simplify ${({(x^{2} + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.4.2.1.1

Multiply the exponents in ${({(x^{2} + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.4.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} + 1)}^{\frac{1}{2} \cdot 2} = {(e^{y} - x)}^{2}$

Step 4.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.4.2.1.1.2.1

Cancel the common factor.

${(x^{2} + 1)}^{\frac{1}{2} \cdot 2} = {(e^{y} - x)}^{2}$

Step 4.4.2.1.1.2.2

Rewrite the expression.

${(x^{2} + 1)}^{1} = {(e^{y} - x)}^{2}$

Step 4.4.2.1.2

Simplify.

$x^{2} + 1 = {(e^{y} - x)}^{2}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Simplify ${(e^{y} - x)}^{2}$ .

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Step 4.4.3.1.1

Rewrite ${(e^{y} - x)}^{2}$ as $(e^{y} - x) (e^{y} - x)$ .

$x^{2} + 1 = (e^{y} - x) (e^{y} - x)$

Step 4.4.3.1.2

Expand $(e^{y} - x) (e^{y} - x)$ using the FOIL Method.

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Step 4.4.3.1.2.1

Apply the distributive property.

$x^{2} + 1 = e^{y} (e^{y} - x) - x (e^{y} - x)$

Step 4.4.3.1.2.2

Apply the distributive property.

$x^{2} + 1 = e^{y} e^{y} + e^{y} (- x) - x (e^{y} - x)$

Step 4.4.3.1.2.3

Apply the distributive property.

$x^{2} + 1 = e^{y} e^{y} + e^{y} (- x) - x e^{y} - x (- x)$

Step 4.4.3.1.3

Simplify and combine like terms.

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Step 4.4.3.1.3.1

Simplify each term.

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Step 4.4.3.1.3.1.1

Multiply $e^{y}$ by $e^{y}$ by adding the exponents.

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Step 4.4.3.1.3.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} + 1 = e^{y + y} + e^{y} (- x) - x e^{y} - x (- x)$

Step 4.4.3.1.3.1.1.2

Add $y$ and $y$ .

$x^{2} + 1 = e^{2 y} + e^{y} (- x) - x e^{y} - x (- x)$

Step 4.4.3.1.3.1.2

Rewrite using the commutative property of multiplication.

$x^{2} + 1 = e^{2 y} - e^{y} x - x e^{y} - x (- x)$

Step 4.4.3.1.3.1.3

Rewrite using the commutative property of multiplication.

$x^{2} + 1 = e^{2 y} - e^{y} x - x e^{y} - 1 \cdot - 1 x \cdot x$

Step 4.4.3.1.3.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 4.4.3.1.3.1.4.1

Move $x$ .

$x^{2} + 1 = e^{2 y} - e^{y} x - x e^{y} - 1 \cdot - 1 (x \cdot x)$

Step 4.4.3.1.3.1.4.2

Multiply $x$ by $x$ .

$x^{2} + 1 = e^{2 y} - e^{y} x - x e^{y} - 1 \cdot - 1 x^{2}$

Step 4.4.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

$x^{2} + 1 = e^{2 y} - e^{y} x - x e^{y} + 1 x^{2}$

Step 4.4.3.1.3.1.6

Multiply $x^{2}$ by $1$ .

$x^{2} + 1 = e^{2 y} - e^{y} x - x e^{y} + x^{2}$

Step 4.4.3.1.3.2

Subtract $x e^{y}$ from $- e^{y} x$ .

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Step 4.4.3.1.3.2.1

Move $e^{y}$ .

$x^{2} + 1 = e^{2 y} - x e^{y} - x e^{y} + x^{2}$

Step 4.4.3.1.3.2.2

Subtract $x e^{y}$ from $- x e^{y}$ .

$x^{2} + 1 = e^{2 y} - 2 x e^{y} + x^{2}$

Step 4.5

Solve for $x$ .

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Step 4.5.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$e^{2 y} - 2 x e^{y} + x^{2} = x^{2} + 1$

Step 4.5.2

Move all terms containing $x$ to the left side of the equation.

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Step 4.5.2.1

Subtract $x^{2}$ from both sides of the equation.

$e^{2 y} - 2 x e^{y} + x^{2} - x^{2} = 1$

Step 4.5.2.2

Combine the opposite terms in $e^{2 y} - 2 x e^{y} + x^{2} - x^{2}$ .

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Step 4.5.2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$e^{2 y} - 2 x e^{y} + 0 = 1$

Step 4.5.2.2.2

Add $e^{2 y} - 2 x e^{y}$ and $0$ .

$e^{2 y} - 2 x e^{y} = 1$

Step 4.5.3

Subtract $e^{2 y}$ from both sides of the equation.

$- 2 x e^{y} = 1 - e^{2 y}$

Step 4.5.4

Divide each term in $- 2 x e^{y} = 1 - e^{2 y}$ by $- 2 e^{y}$ and simplify.

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Step 4.5.4.1

Divide each term in $- 2 x e^{y} = 1 - e^{2 y}$ by $- 2 e^{y}$ .

$\frac{- 2 x e^{y}}{- 2 e^{y}} = \frac{1}{- 2 e^{y}} + \frac{- e^{2 y}}{- 2 e^{y}}$

Step 4.5.4.2

Simplify the left side.

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Step 4.5.4.2.1

Cancel the common factor of $- 2$ .

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Step 4.5.4.2.1.1

Cancel the common factor.

$\frac{- 2 x e^{y}}{- 2 e^{y}} = \frac{1}{- 2 e^{y}} + \frac{- e^{2 y}}{- 2 e^{y}}$

Step 4.5.4.2.1.2

Rewrite the expression.

$\frac{x e^{y}}{e^{y}} = \frac{1}{- 2 e^{y}} + \frac{- e^{2 y}}{- 2 e^{y}}$

Step 4.5.4.2.2

Cancel the common factor of $e^{y}$ .

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Step 4.5.4.2.2.1

Cancel the common factor.

$\frac{x e^{y}}{e^{y}} = \frac{1}{- 2 e^{y}} + \frac{- e^{2 y}}{- 2 e^{y}}$

Step 4.5.4.2.2.2

Divide $x$ by $1$ .

$x = \frac{1}{- 2 e^{y}} + \frac{- e^{2 y}}{- 2 e^{y}}$

Step 4.5.4.3

Simplify the right side.

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Step 4.5.4.3.1

Simplify each term.

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Step 4.5.4.3.1.1

Move the negative in front of the fraction.

$x = - \frac{1}{2 e^{y}} + \frac{- e^{2 y}}{- 2 e^{y}}$

Step 4.5.4.3.1.2

Cancel the common factor of $e^{2 y}$ and $e^{y}$ .

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Step 4.5.4.3.1.2.1

Factor $e^{y}$ out of $- e^{2 y}$ .

$x = - \frac{1}{2 e^{y}} + \frac{e^{y} (- e^{y})}{- 2 e^{y}}$

Step 4.5.4.3.1.2.2

Cancel the common factors.

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Step 4.5.4.3.1.2.2.1

Factor $e^{y}$ out of $- 2 e^{y}$ .

$x = - \frac{1}{2 e^{y}} + \frac{e^{y} (- e^{y})}{e^{y} \cdot - 2}$

Step 4.5.4.3.1.2.2.2

Cancel the common factor.

$x = - \frac{1}{2 e^{y}} + \frac{e^{y} (- e^{y})}{e^{y} \cdot - 2}$

Step 4.5.4.3.1.2.2.3

Rewrite the expression.

$x = - \frac{1}{2 e^{y}} + \frac{- e^{y}}{- 2}$

Step 4.5.4.3.1.3

Dividing two negative values results in a positive value.

$x = - \frac{1}{2 e^{y}} + \frac{e^{y}}{2}$